Here we should invoke the definitions of [1]}. Suppose \({X}, {Y}\)  are vector spaces and \(T : {X} \rightarrow {Y}\)  is a linear transformation. Let \({Y}^{\prime } := \text{all of the linear functionals of ${Y} \rightarrow \mathbb {F}$}\) .
Note this \({Y}^{\prime }\)  is not necessarily equivalent to \({Y}^*\)  because we didn't impose that \({X}, {Y}\)  are Banach / Hilbert spaces.
If \(y^{\prime } \in {Y}^{\prime }\) , then \(y^{\prime } \circ T : {X} \rightarrow \mathbb {F}\)  is easily seen to be a linear functional on \({X}\) . That is, \(y^{\prime } \circ T \in {X}^{\prime }\) . This defines a map,
\(T^{\prime } : {Y}^{\prime } \rightarrow {X}^{\prime }\) 
